Annotation of OpenXM/src/asir-contrib/packages/doc/n_wishartd/n_wishartd-en.texi, Revision 1.2
1.2 ! noro 1: %comment $OpenXM: OpenXM/src/asir-contrib/packages/doc/n_wishartd/n_wishartd-en.texi,v 1.1 2016/08/24 08:10:37 noro Exp $
1.1 noro 2: %comment --- おまじない ---
3: \input ../../../../asir-doc/texinfo
4: @iftex
5: @catcode`@#=6
6: @def@fref#1{@xrefX[#1,,@code{#1},,,]}
7: @def@b#1{{@bf@gt #1}}
8: @catcode`@#=@other
9: @end iftex
10: @overfullrule=0pt
11: @c -*-texinfo-*-
12: @comment %**start of header
13: @comment --- おまじない終り ---
14:
15: @comment --- GNU info ファイルの名前 ---
16: @setfilename asir-contrib-n_wishartd
17:
18: @comment --- タイトル ---
19: @settitle n_wishartd
20:
21: @comment %**end of header
22: @comment %@setchapternewpage odd
23:
24: @comment --- おまじない ---
25: @ifinfo
26: @macro fref{name}
27: @ref{\name\,,@code{\name\}}
28: @end macro
29: @end ifinfo
30:
31: @iftex
32: @comment @finalout
33: @end iftex
34:
35: @titlepage
36: @comment --- おまじない終り ---
37:
38: @comment --- タイトル, バージョン, 著者名, 著作権表示 ---
39: @title n_wishartd
40: @subtitle n_wishartd User's Manual
41: @subtitle Edition 1.0
42: @subtitle Aug 2016
43:
44: @author by Masayuki Noro
45: @page
46: @vskip 0pt plus 1filll
47: Copyright @copyright{} Masayuki Noro
48: 2016. All rights reserved.
49: @end titlepage
50:
51: @comment --- おまじない ---
52: @synindex vr fn
53: @comment --- おまじない終り ---
54:
55: @comment --- @node は GNU info, HTML 用 ---
56: @comment --- @node の引数は node-name, next, previous, up ---
57: @node Top,, (dir), (dir)
58:
59: @comment --- @menu は GNU info, HTML 用 ---
60: @comment --- chapter 名を正確に並べる ---
61: @menu
62: * n_wishartd.rr ::
63: * Index::
64: @end menu
65:
66: @comment --- chapter の開始 ---
67: @comment --- 親 chapter 名を正確に ---
68: @node n_wishartd.rr ,,, Top
69: @chapter n_wishartd.rr
70: @comment --- section 名を正確に並べる ---
71: @menu
72: * Restrction of matrix 1F1 on diagonal regions::
73: * Numerical comptation of restricted function::
74: * Differential operators with partial fraction coefficients::
75: * Experimental implementation of Runge-Kutta methods::
76: @end menu
77:
78: This manual explains about @samp{n_wishartd.rr},
79: a package for computing a system of differential equations
80: satisfied by the matrix 1F1 on a diagonal region.
81: To use this package one has to load @samp{n_wishartd.rr}.
82: @example
83: [...] load("n_wishartd.rr");
84: @end example
85: @noindent
86: A prefix @code{n_wishartd.} is necessary to call the functions in this package.
87:
88: @comment --- section の開始 ---
89: @comment --- 書体指定について ---
90: @comment --- @code{} はタイプライタ体表示 ---
91: @comment --- @var{} は斜字体表示 ---
92: @comment --- @b{} はボールド表示 ---
93: @comment --- @samp{} はファイル名などの表示 ---
94:
95: @node Restriction of matrix 1F1 on diagonal regions,,, n_wishartd.rr
96: @section Restriction of matrix 1F1 on diagonal regions
97:
98: @menu
99: * n_wishartd.diagpf::
100: * n_wishartd.message::
101: @end menu
102:
103: @node n_wishartd.n_wishartd.diagpf,,, Restriction of matrix 1F1 on diagonal regions
104:
105: @subsection @code{n_wishartd.diagpf}
106: @findex n_wishartd.diagpf
107:
108: @table @t
109: @item n_wishartd.diagpf(@var{m},@var{blocks})
110: computes a system of PDEs satisfied by the @var{m} variate matrix 1F1 on a diagonal region specified
111: by @var{blocks}.
112: @end table
113:
114: @table @var
115: @item return
116: A list @var{[E1,E2,...]} where each @var{Ei} is
117: a differential operator with partial fraction coefficients and it annihilates the restricted 1F1.
118:
119: @item m
120: A natural number
121: @item vars
122: A list @var{[[s1,e1],[s2,e2],...]}.
123: @item options
124: See below.
125: @end table
126:
127: @itemize @bullet
128: @item This function computes a system of PDEs satisfied by the @var{m} variate matrix 1F1 on a diagonal region specified by @var{blocks}.
129: @item Each component @var{[s,e]} in @var{blocks} denotes @var{ys=y(s+1)=...=ye}. The representative variable of this block is @var{ye}.
130: @item One has to specify @var{blocks} so that all the variables appear in it. In particular a block which contains only one variable
131: is specified by @var{[s,s]}.
132: @item It has not yet been proven that this function always succeeds. At least it is known that this function succeeds if the size of each block <= 36.
133: @item See @ref{Differential operators with partial fraction coefficients} for the format of the result.
134: @end itemize
135:
136: @example
137: [2649] Z=n_wishartd.diagpf(5,[[1,3],[4,5]]);
138: [
139: [[[[-1,[]]],(1)*<<0,0,0,0,3,0>>],
140: [[[-2,[[y1-y4,1]]],[-2,[[y4,1]]]],(1)*<<0,1,0,0,1,0>>],
141: [[[9/2,[[y1-y4,1]]],[-3*c+11/2,[[y4,1]]],[3,[]]],(1)*<<0,0,0,0,2,0>>],
142: ...
143: [[[-6*a,[[y1-y4,1],[y4,1]]],[(4*c-10)*a,[[y4,2]]],[-4*a,[[y4,1]]]],
144: (1)*<<0,0,0,0,0,0>>]],
145: [[[[-1,[]]],(1)*<<0,4,0,0,0,0>>],
146:
147: [[[-6,[[y1-y4,1]]],[-6*c+10,[[y1,1]]],[6,[]]],(1)*<<0,3,0,0,0,0>>],
148: [[[5,[[y1-y4,1]]],[-5,[[y1,1]]]],(1)*<<0,2,0,0,1,0>>],
149: ...
150: [[[21*a,[[y1-y4,2],[y1,1]]],[(36*c-87)*a,[[y1-y4,1],[y1,2]]],
151: [-36*a,[[y1-y4,1],[y1,1]]],[(18*c^2-84*c+96)*a,[[y1,3]]],
152: [-9*a^2+(-36*c+87)*a,[[y1,2]]],[18*a,[[y1,1]]]],(1)*<<0,0,0,0,0,0>>]]
153: ]
154: @end example
155:
156: @node n_wishartd.message,,, Restriction of matrix 1F1 on diagonal regions
157:
158: @subsection @code{n_wishartd.message}
159: @findex n_wishartd.message
160:
161: @table @t
162: @item n_wishartd.message(@var{onoff})
1.2 ! noro 163: starts/stops displaying messages during computation.
1.1 noro 164: @end table
165:
166: @table @var
167: @item onoff
1.2 ! noro 168: Start displaying messages if @var{onoff}=1.
! 169: Stop displaying messages if @var{onoff}=0.
1.1 noro 170: @end table
171:
172: @itemize @bullet
1.2 ! noro 173: @item This function starts/stops displaying messages during computation.
! 174: The default value is set to 0.
1.1 noro 175: @end itemize
176:
177: @node Numerical comptation of restricted function,,, n_wishartd.rr
178: @section Numerical comptation of restricted function
179:
180: @menu
181: * n_wishartd.prob_by_hgm::
182: * n_wishartd.prob_by_ps::
183: * n_wishartd.ps::
184: @end menu
185:
186: @node n_wishartd.prob_by_hgm,,, Numerical comptation of restricted function
187: @subsection @code{n_wishartd.prob_by_hgm}
188: @findex n_wishartd.prob_by_hgm
189:
190: @table @t
1.2 ! noro 191: @item n_wishartd.prob_by_hgm(@var{m},@var{n},@var{[p1,p2,...]},@var{[s1,s2,...]},@var{t}[|@var{options}])
! 192: computes the value of the distribution function of the largest eigenvalue of a Wishart matrix.
1.1 noro 193: @end table
194:
195: @table @var
196: @item return
197: @item m
1.2 ! noro 198: The number of variables.
1.1 noro 199: @item n
1.2 ! noro 200: The degrees of freedom.
! 201: @item [p1,p2,...,pk]
! 202: A list of the multiplicities of repeated eigenvalues.
! 203: @item [s1,s2,...,sk]
! 204: A list of repeated eigenvalues.
1.1 noro 205: @end table
206:
207: @itemize @bullet
208: @item
1.2 ! noro 209: Let @var{l1} be the largest eigenvalue of a Wishart matrix.
! 210: Let @var{Pr[l1<t]} be the distribution function of @var{l1}.
! 211: The function @code{n_wishartd.prob_by_hgm} computes the value of the distribution function by using HGM
! 212: for a covariance matrix which has repeated eigenvalues @var{si} with multiplicity @var{pi} (@var{i=1,...,k}).
! 213:
! 214: @item This function repeats a Runge-Kutta method for the Pfaffian system by doubling the step size
! 215: until the relative error between the current result and the previous result is less than @var{eps},
! 216: The default value of @var{eps} is @var{10^(-4)}.
1.1 noro 217: @item
1.2 ! noro 218: If an option @var{eq} is not set,
! 219: a system of PDES satisfied by 1F1 on the diagonal region specified by @var{[p1,p2,...]} is computed.
! 220: If an option @var{eq} is set, the list specified by @var{eq} is regarded as the correct system of PDEs.
! 221: @item If an option @var{eps} is set, the value is used as @var{eps}.
! 222: @item If an option @var{td} is set, the truncated power series solution for computing the initial vector
! 223: is computed up to the total degree specified by @var{td}. The default value is 100.
! 224: @item If an option @var{rk} is set, it is regarded as the order of a Runge-Kutta method. The default vaule is 5.
! 225: @item It is recommended to use this function only when @var{k<=2} where @var{k} is the number of diagonal blocks because of
! 226: the difficulty of the truncated power series solution and the difficulty of computation of the Pfaffian matrices.
1.1 noro 227: @end itemize
228:
229: @example
230: [...] n_wishartd.message(1)$
231: [...] P=n_wishartd.prob_by_hgm(10,100,[9,1],[1/100,1],100|eps=10^(-6));
232: ...
233: [x0=,8/25]
234: Step=10000
235: [0]
236: [8.23700622458446e-17,8.23700622459772e-17]
237: ...
238: Step=1280000
239: [0][100000][200000][300000]...[900000][1000000][1100000][1200000]
240: [0.516246820120598,0.516246820227214]
241: [log ratio=,4.84611265040128]
242:
243: Step=2560000
244: [0][100000][200000][300000]...[2200000][2300000][2400000][2500000]
245: [0.516246912003845,0.516246912217004]
246: [log ratio=,4.93705929488356]
247: [diag,18.6292,pfaffian,1.09207,ps,41.0026,rk,213.929]
248: 0.516246912217004
249: 266.4sec + gc : 8.277sec(276.8sec)
250: @end example
251:
252: @node n_wishartd.prob_by_ps,,, Numerical comptation of restricted function
253: @subsection @code{n_wishartd.prob_by_ps}
254: @findex n_wishartd.prob_by_ps
255:
256: @table @t
257: @item n_wishartd.prrob_by_ps(@var{m},@var{n},@var{[p1,p2,...]},@var{[s1,s2,...]},@var{t}[|@var{options}])
1.2 ! noro 258: computes the value of the distribution function of the largest eigenvalue of a Wishart matrix.
1.1 noro 259: @end table
260:
261: @table @var
1.2 ! noro 262: @item return
1.1 noro 263: @item m
1.2 ! noro 264: The number of variables.
1.1 noro 265: @item n
1.2 ! noro 266: The degrees of freedom.
! 267: @item [p1,p2,...,pk]
! 268: A list of the multiplicities of repeated eigenvalues.
! 269: @item [s1,s2,...,sk]
! 270: A list of repeated eigenvalues.
1.1 noro 271: @end table
272:
273: @itemize @bullet
274: @item
1.2 ! noro 275: This function compute a truncated power series solution up to a total degree
! 276: where the relative error between the current value and the previous value at the desired point
! 277: is less than @var{eps}. The default value of @var{eps} is @var{10^(-4)}.
! 278: The value of the distribution function is computed by using this power series.
! 279: @item If an option @var{eps} is set, the value is used as @var{eps}.
! 280: @item
! 281: If an option @var{eq} is not set,
! 282: a system of PDES satisfied by 1F1 on the diagonal region specified by @var{[p1,p2,...]} is computed.
! 283: If an option @var{eq} is set, the list specified by @var{eq} is regarded as the correct system of PDEs.
! 284: @item It is recommened to use this function when @var{t} is small.
1.1 noro 285: @end itemize
286:
287: @example
288: [...] Q=n_wishartd.prob_by_ps(10,100,[9,1],[1/100,1],1/2);
289: ...
290: [I=,109,act,24.9016,actmul,0,gr,19.7852]
291: 2.69026137621748e-165
292: 61.69sec + gc : 2.06sec(64.23sec)
293: [...] R=n_wishartd.prob_by_hgm(10,100,[9,1],[1/100,1],1/2|td=50);
294: [diag,15.957,pfaffian,1.00006,ps,5.92437,rk,1.29208]
295: 2.69026135182769e-165
296: 23.07sec + gc : 1.136sec(24.25sec)
297: @end example
298:
299: @node n_wishartd.ps,,, Numerical comptation of restricted function
300: @subsection @code{n_wishartd.ps}
301: @findex n_wishartd.ps
302:
303: @table @t
304: @item n_wishartd.ps(@var{z},@var{v},@var{td})
1.2 ! noro 305: computes a truncated power series solution up to the total degree @var{td}.
1.1 noro 306: @end table
307:
308: @table @var
309: @item return
1.2 ! noro 310: A list of polynomial
1.1 noro 311: @item z
1.2 ! noro 312: A list of differential operators with partial fraction coefficients.
1.1 noro 313: @item v
1.2 ! noro 314: A list of variables.
1.1 noro 315: @end table
316:
317: @itemize @bullet
318: @item
1.2 ! noro 319: The result is a list @var{[p,pd]} where
! 320: @var{p} is a truncated power series solution up to the total degree @var{td}
! 321: and @var{pd} is the @var{td} homogeneous part of @var{p}.
! 322: @item @var{z} cannot contain parameters other than the variables in @var{v}.
1.1 noro 323: @end itemize
324:
325: @example
326: [...] Z=n_wishartd.diagpf(10,[[1,5],[6,10]])$
327: [...] Z0=subst(Z,a,(10+1)/2,c,(10+100+1)/2)$
328: [...] PS=n_wishartd.ps(Z0,[y1,y6],10)$
329: [...] PS[0];
330: 197230789502743383953639/230438384724900975787223158176000*y1^10+
331: ...
332: +(6738842542131976871672233/1011953706634779427957034268904320*y6^9
333: ...+3932525/62890602*y6^2+1025/4181*y6+55/111)*y1
334: +197230789502743383953639/230438384724900975787223158176000*y6^10
335: +...+1395815/62890602*y6^3+3175/25086*y6^2+55/111*y6+1
336: @end example
337:
338: @node Differential operators with partial fraction coefficients,,, n_wishartd.rr
339: @section Differential operators with partial fraction coefficients
340:
341: @menu
1.2 ! noro 342: * Representation of partial fractions::
! 343: * Representation of differential operators with partial fraction coefficients::
1.1 noro 344: * Operations on differential operators with partial fraction coefficients::
345: @end menu
346:
1.2 ! noro 347: @node Representation of partial fractions,,, Differential operators with partial fraction coefficients
! 348: @subsection Representation of partial fractions
1.1 noro 349:
1.2 ! noro 350: The coefficients of the PDE satisfied by the matrix 1F1 are written
! 351: as a sum of @var{1/yi} and @var{1/(yi-yj)} multiplied by constants.
! 352: Furthermore the result of diagonalization by l'Hopital rule
! 353: can also be written as a sum of partial fractions.
1.1 noro 354:
355: @itemize @bullet
356: @item
1.2 ! noro 357: A product @var{yi0^n0(yi1-yj1)^n1(yi2-yj2)^n2...(yik-yjk)^nk}
! 358: in the denominator of a fraction is represented as a list @var{[[yi0,n0],[yi1-yj1,n1],...,[yik-yjk,nk]]},
! 359: Where each @var{yi-yj} satisfies @var{i>j} and the factors are sorted according to an ordering.
1.1 noro 360: @item
1.2 ! noro 361: Let @var{f} be a power sum as above and @var{c} a constant.
! 362: Then a monomial @var{c/f} is represented by a list は @var{[c,f]}.
! 363: @var{f=[]} means that the denominator is 1.
1.1 noro 364: @item
1.2 ! noro 365: Finally @var{c1/f1+...+ck/fk} is represented as a list @var{[[c1,f1],...,[ck,fk]]},
! 366: where terms are sorted according to an ordering.
1.1 noro 367: @item
1.2 ! noro 368: We note that it is possible that a partial fraction is reduced to 0.
1.1 noro 369: @end itemize
370:
1.2 ! noro 371: @node Representation of differential operators with partial fraction coefficients,,, Differential operators with partial fraction coefficients
! 372: @subsection Representation of differential operators with partial fraction coefficients
1.1 noro 373:
1.2 ! noro 374: By using partial fractions explained in the previous section,
! 375: differential operators with partial fraction coefficients are represented.
! 376: Let @var{f1,...,fk} be partial fractions and @var{d1,...,dk} distributed monomials such that
! 377: @var{d1>...>dk}) with respected to the current monomial ordering.
! 378: Then a differential operator @var{f1*d1+...+fk*dk} is represented as a list @var{[f1,d1],...[fk,dk]]}.
1.1 noro 379:
380: @node Operations on differential operators with partial fraction coefficients,,, Differential operators with partial fraction coefficients
381: @subsection Operations on differential operators with partial fraction coefficients
382:
383: @menu
384: * n_wishartd.wsetup::
385: * n_wishartd.addpf::
386: * n_wishartd.mulcpf::
387: * n_wishartd.mulpf::
388: * n_wishartd.muldpf::
389: @end menu
390:
391: @node n_wishartd.wsetup,,, Operations on differential operators with partial fraction coefficients
392: @subsubsection @code{n_wishartd.wsetup}
393: @findex n_wishartd.wsetup
394:
395: @table @t
396: @item n_wishartd.wsetup(@var{m})
397: @end table
398:
399: @table @var
400: @item m
1.2 ! noro 401: A natural number.
1.1 noro 402: @end table
403:
404: @itemize @bullet
1.2 ! noro 405: @item This function sets a @var{m}-variate computational enviroment. The variables are @var{y0,y1,...,ym} and @var{dy0,...,dym},
! 406: where @var{y0, dy0} are dummy variables for intermediate computation.
1.1 noro 407: @end itemize
408:
409: @node n_wishartd.addpf,,, Operations on differential operators with partial fraction coefficients
410: @subsubsection @code{n_wishartd.addpf}
411: @findex n_wishartd.addpf
412: @table @t
413: @item n_wishartd.addpf(@var{p1},@var{p2})
414: @end table
415:
416: @table @var
417: @item return
1.2 ! noro 418: A differential operator with partial fraction coefficients.
1.1 noro 419: @item p1, p2
1.2 ! noro 420: Differential operators with partial fraction coefficients.
1.1 noro 421: @end table
422:
423: @itemize @bullet
1.2 ! noro 424: @item This function computes the sum of differential operators @var{p1} and @var{p2}.
1.1 noro 425: @end itemize
426:
427: @node n_wishartd.mulcpf,,, Operations on differential operators with partial fraction coefficients
428: @subsubsection @code{n_wishartd.mulcpf}
429: @findex n_wishartd.mulcpf
430: @table @t
431: @item n_wishartd.mulcpf(@var{c},@var{p})
432: @end table
433:
434: @table @var
435: @item return
1.2 ! noro 436: A differential operator with partial fraction coefficients.
1.1 noro 437: @item c
1.2 ! noro 438: A partial fraction.
1.1 noro 439: @item p
1.2 ! noro 440: Differential operators with partial fraction coefficients.
1.1 noro 441: @end table
442:
443: @itemize @bullet
1.2 ! noro 444: @item This function computes the product of a partial fraction @var{c} and a differential operator @var{p}.
1.1 noro 445: @end itemize
446:
447: @node n_wishartd.mulpf,,, Operations on differential operators with partial fraction coefficients
448: @subsubsection @code{n_wishartd.mulpf}
449: @findex n_wishartd.mulpf
450: @table @t
451: @item n_wishartd.mulpf(@var{p1},@var{p2})
452: @end table
453:
454: @table @var
455: @item return
1.2 ! noro 456: A differential operator with partial fraction coefficients.
1.1 noro 457: @item p1, p2
1.2 ! noro 458: Differential operators with partial fraction coefficients.
1.1 noro 459: @end table
460:
461: @itemize @bullet
1.2 ! noro 462: @item This function computes the product of differential operators @var{p1} and @var{p2}.
1.1 noro 463: @end itemize
464:
465: @node n_wishartd.muldpf,,, Operations on differential operators with partial fraction coefficients
466: @subsubsection @code{n_wishartd.muldpf}
467: @findex n_wishartd.muldpf
468: @table @t
469: @item n_wishartd.muldpf(@var{y},@var{p})
470: @end table
471:
472: @table @var
473: @item return
1.2 ! noro 474: A differential operator with partial fraction coefficients.
1.1 noro 475: @item y
1.2 ! noro 476: A variable.
1.1 noro 477: @item p
1.2 ! noro 478: A differential operator with partial fraction coefficients.
1.1 noro 479: @end table
480:
481: @itemize @bullet
1.2 ! noro 482: @item
! 483: This function computes the product of the differential operator @var{dy} corresponding to a variable @var{y} and @var{p}.
1.1 noro 484: @end itemize
485:
486: @example
487: [...] n_wishartd.wsetup(4)$
488: [...] P=n_wishartd.wishartpf(4,1);
489: [[[[1,[]]],(1)*<<0,2,0,0,0>>],[[[1/2,[[y1-y2,1]]],[1/2,[[y1-y3,1]]],
490: ...,[[[-a,[[y1,1]]]],(1)*<<0,0,0,0,0>>]]
491: [...] Q=n_wishartd.muldpf(y1,P);
492: [[[[1,[]]],(1)*<<0,3,0,0,0>>],[[[1/2,[[y1-y2,1]]],[1/2,[[y1-y3,1]]],
493: ...,[[[a,[[y1,2]]]],(1)*<<0,0,0,0,0>>]]
494: @end example
495:
496: @node Experimental implementation of Runge-Kutta methods ,,, n_wishartd.rr
497: @section Experimental implementation of Runge-Kutta methods
498:
499: @menu
500: * rk_ratmat::
501: @end menu
502:
503: @node rk_ratmat,,, Experimental implementation of Runge-Kutta methods
504:
1.2 ! noro 505: In the function @code{n_wishartd.ps_by_hgm}, after computing the Pfaffian matrices for
! 506: the sytem of PDEs on a diagonal region, it executes a built-in function
! 507: @code{rk_ratmat} which computes an approximate solution of the Pfaffian system
! 508: by Runge-Kutta method for a spcified step size.
! 509: This function is repeated until the result gets stabilized, by doubling the step size.
! 510: @code{rk_ratmat} can be used as a general-purpose Runge-Kutta driver and we explain how to use it.
1.1 noro 511:
512: @subsection @code{rk_ratmat}
513: @findex rk_ratmat
514:
515: @table @t
516: @item rk_ratmat(@var{rk45},@var{num},@var{den},@var{x0},@var{x1},@var{s},@var{f0})
1.2 ! noro 517: solves a system of linear ODEs with rational function coefficients.
1.1 noro 518: @end table
519:
520: @table @var
521: @item return
1.2 ! noro 522: A list of real numbers.
1.1 noro 523: @item rk45
1.2 ! noro 524: 4 or 5.
1.1 noro 525: @item num
1.2 ! noro 526: An array of constant matrices.
1.1 noro 527: @item den
1.2 ! noro 528: A polynomial.
1.1 noro 529: @item x0, x1
1.2 ! noro 530: Real numbers.
1.1 noro 531: @item s
1.2 ! noro 532: A natural number.
1.1 noro 533: @item f0
1.2 ! noro 534: A real vector.
1.1 noro 535: @end table
536:
537: @itemize @bullet
538: @item
1.2 ! noro 539: Let @var{k} be the size of an array @var{num}.
! 540: The function @code{rk_ratmat} solves an initial value problem
! 541: @var{dF/dx = P(x)F}, @var{F(x0)=f0} for @var{P(x)=1/den(num[0]+num[1]x+...+num[k-1]x^(k-1))} by a Runge-Kutta method.
1.1 noro 542: @item
1.2 ! noro 543: @var{rk45} specifies the order of a Runge-Kutta method. Adaptive methods are not implemented.
1.1 noro 544: @item
1.2 ! noro 545: The step size is specified by @var{s}. The step width is @var{(x1-x0)/s}.
1.1 noro 546: @item
1.2 ! noro 547: If the size of @var{f0} is @var{n}, each component of @var{num} is a square matrix of size @var{n}.
1.1 noro 548: @item
1.2 ! noro 549: The result is a list of real numbers @var{[r1,...,rs]} of length @var{s}.
! 550: @var{ri} is the 0-th component of the solution vector after the step @var{i}.
! 551: Before going to the next step the solution vector is divided by @var{ri}.
! 552: Therefore the 0-th component of the final solution vector [var{F(x1)} is equal to @var{rs*r(s-1)*...*r1}.
! 553: @item Since the ODE is linear, each step of Runge-Kutta method is also linear.
! 554: This enables us to apply a normalization such that the 0-th
! 555: component of each intermediate solution vector is set to 1. By
! 556: applying this normalization we expect that all the components of
! 557: intermediate solution vectors can be represented by the format of
! 558: double precision floating point number.
! 559: If there exist some components in the initial vector @var{f0}, we apply this normalization
! 560: to @var{f0}. After applying @code{rk_ratmat} we multiply the result for the normalized @var{f0} and the 0-th component
! 561: of the original @var{f0} to get the desired result.
1.1 noro 562: @end itemize
563:
564: @example
565: [...] F=ltov([sin(1/x),cos(1/x),sin(1/x^2),cos(1/x^2)]);
566: [ sin((1)/(x)) cos((1)/(x)) sin((1)/(x^2)) cos((1)/(x^2)) ]
567: [...] F0=map(eval,map(subst,F,x,1/10));
568: [ -0.54402111088937 -0.839071529076452 -0.506365641109759 0.862318872287684 ]
569: [...] N0=matrix(4,4,[[0,0,0,0],[0,0,0,0],[0,0,0,-2],[0,0,2,0]])$
570: [...] N1=matrix(4,4,[[0,-1,0,0],[1,0,0,0],[0,0,0,0],[0,0,0,0]])$
571: [...] N=ltov([N0,N1])$
572: [...] D=x^3$
573: [...] R=rk_ratmat(5,N,D,1/10,10,10^4,F0)$
574: [...] for(T=R,A=1;T!=[];T=cdr(T))A *=car(T)[1];
575: [...] A;
576: 0.0998334
577: [...] F1=map(eval,map(subst,F,x,10));
578: [ 0.0998334166468282 0.995004165278026 0.00999983333416666 0.999950000416665 ]
579: @end example
580:
581:
582: @comment --- おまじない ---
583: @node Index,,, Top
584: @unnumbered Index
585: @printindex fn
586: @printindex cp
587: @iftex
588: @vfill @eject
589: @end iftex
590: @summarycontents
591: @contents
592: @bye
593: @comment --- おまじない終り ---
594:
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